If fuzzy logic is construed, as Zadeh and co. suggest it should be, as a nonclassical theory of truth-preserving inferences, fuzzy technology does not rely on it, and so the successes of that technology cannot be claimed to its credit. If, on the other hand, fuzzy logic is construed as an attempt to represent the mental processes through which people go when making adjustments to kiln thermostats, air-conditionaers, etc., there is a connection with fuzzy technology. But, of course, so construed, fuzzy logic is not, after all, an attempt to represent truth-preserving inferences, and is not, after all, a theory in the same domain as classical logic; in fact, so construed, it is obviously not properly describable as a "logic" at all.

Susan Haack, from Deviant Logic, Fuzzy Logic, 1996.

756

While, traditionally, logic has corrected or avoided it, fuzzy logic compromises with vagueness; it is not just a logic of vagueness, it is -- from what Frege"s point of view would have been a contradiction in terms -- a vague logic.

Susan Haack, from Deviant Logic, Fuzzy Logic, 1996.

755

Fuzzy logic lacks every feature that the pioneers of modern logic wanted logic for; it sacrifices what have traditionally been regarded as the crucuail advantages of formalism -- precise, formal rules of inference, the security offered by consistentcy and completeness results.

Susan Haack, from Deviant Logic, Fuzzy Logic, 1996.

754

The shortest path between two assertions about the reals passes through the
complexes.

Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.

Jacques Hadamard, quoted in Euclidean and Non-Euclidean Geometries by Greenberg

875

Anyone, anywhere along the line, can fill in the details and check them. The fact that a computer can run through more details in a few hours than a human could ever hope to do in a lifetime does not change the basic concept of mathematical proof. What has changed is not the theory but the practice of mathematics.

Wofgang Haken, quoted in From Here to Infinity, by Ian Stewart.

585

The universe is not only queerer than we suppose but queerer than we can suppose.

J.B.S. Haldane, quoted in Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, by Manfred Schroder.

566

The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method, the method in which the teacher plays a role of an omniscient but largely uncommunicative referee between the learner and the facts.

Paul Halmos, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz

450

It's hard for me to get used to the absence of pressure. I always pushed myself under pressure, and of course, I blamed the world. The world is putting on the pressure. Well, now I'm beginning to realize that the world is not putting on the pressure. If I never published anything, not even an elementary textbook, if I never again answered a letter, if I never did anything any more except drink my beer and watch the telly, nobody would, I think, think any the worse of me. But I keep putting myself under a little pressure and keep doing these small piddling jobs.

Learning mathematics is always extraordinarily hard work. I can't easily read mathematics. I can't listen to lectures. The only thing I enjoy is a kind of mathematical gossip, when people sit in easy chairs with their feet up on something and tell me their mathematics; then I can learn.

Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

Paul R. Halmos, from I Want to Be a Mathematician

1148

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

Paul Halmos, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz

451

A smooth lecture... may be pleasant; a good teacher challenges, asks,
irritates and maintains high standards - all that is generally not
pleasant.

Paul Halmos,

151

Teachers of elementary mathematics in the USA frequently complain that all
calculus books are bad. That is a case in point. Calculus books are bad
because there is no such subject as calculus; it is not a subject because
it is many subjects. What we call calculus nowadays is the union of a dab
of logic and set theory, some axiomatic theory of complete ordered fields,
analytic geometry and topology, the latter in both the "general" sense
(limits and continuous functions) and the algebraic sense (orientation),
real-variable theory properly so called (differentiation), the combinatoric
symbol manipulation called formal integration, the first steps of
low-dimensional measure theory, some differential geometry, the first steps
of the classical analysis of the trigonometric, exponential, and
logarithmic functions, and, depending on the space available and the
personal inclination of the author, some cook-book differential equations,
elementary mechanics, and a small assortment of applied mathematics. Any
one of these is hard to write a good book on; the mixture is impossible.

Paul Halmos, quoted in Excursions in Calculus, by Robert Young.

152

Mathematics - this may surprise or shock some - is never deductive in its
creation. The mathematician at work makes vague guesses, visualizes broad
generalizations, and jumps to unwarranted conclusions. He arranges and
rearranges his ideas, and he becomes convinced of their truth long before
he can write down a logical proof... The deductive stage, writing the
result down, and writing its rigorous proof are relatively trivial once the
real insight arrives; it is more like the draftsman's work not the
architect's.

Paul Halmos, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.

150

A good stack of examples, as large as possible, is indispensable for a
thorough understanding of any concept, and when I want to learn something
new, I make it my first job to build one.

Paul Halmos, quoted in Contemporary Abstract Algebra, by J. Gallian.

149

To be able to be caught up into the world of thought - that is educated.

Edith Hamilton, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.

153

This above all: To thine own self be true, And it must follow, as the night the day, Thou canst not then be false to any man.

Hamlet,

936

When the morning's freshness has been replaced by the weariness of midday, when the leg muscles quiver under the strain, the climb seems endless, and suddenly nothing will go quite as you wish - it is then that you must not hesitate.

People usually consider walking on water or in thin air a miracle. But I think the real miracle is not to walk either on water or in thin air, but to walk on earth. Every day we are engaged in a miracle which we don't even recognize: a blue sky, white clouds, green leaves, the black, curious eyes of a child -- our own two eyes. All is a miracle.

Thich Nhat Hanh,

1018

Paul Erdos has a theory that God has a book containing all the theorems of
mathematics with their absolutely most beautiful proofs, and when he wants
to express particular appreciation of a proof he exclaims, 'This is from
the book!'

Ross Hansberger, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.

155

Hardy's New Year's Resolutions:
1. Prove the Riemann hypothesis.
2. Make 211 no out in the fourth innings of the last test match at the Oval.
3. Find an argument for the nonexistence of God which shall convince the general public.
4. Be the first man at the top of Mt. Everest.
5. Be proclaimed the first president of the U.S.S.R., of Great Britian, and Germany.
6. Murder Mussolini.
3. Find an argument for the nonexistence of God which shall convince the general public.
4. Be the first man at the top of Mt. Everest.
5. Be proclaimed the first president of the U.S.S.R., of Great Britian, and Germany.
6. Murder Mussolini.

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel

882

Here was a man who could work out modular equations and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, on the formal side at any rate, beyond that of any mathematician in the world, who had found for himself the functional equation of the Zeta-function, and the dominant terms of many of the most famous problems in the analytic theory of numbers; and he had never heard of a doubly periodic function or of Cauchy's theorem, and he had indeed but the vaguest idea of what a function of a complex variable was. His ideas as to what constituted a mathematical proof were of the most shadowy description. All his results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account.

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel

880

I have [very rarely] encountered a pupil who could face the simplest problem involving the ideas of infinity, limit, or continuity with a vestige of the confidence with which he could deal with questions of a different character and of far greater intrinsic difficulty.

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel

879

He [Ramanujan] would probably have been a greater mathematician if he had been caught and tamed a little in his youth; he would have discovered more that was new, and that, no doubt, of greater importance. On the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain.
He [Ramanujan] would probably have been a greater mathematician if he had been caught and tamed a little in his youth; he would have discovered more that was new, and that, no doubt, of greater importance. On the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain.

G.H. Hardy, from The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel

883

The positive integers stand there, a continual and inevitable challenge to the curiosity of every healthy mind.

G.H. Hardy, quoted in Elementary Number Theory by David M. Burton.

905

I think that it is time that teachers of geometry become a little more ambitious… It seems to me regrettable that students are not given the opportunity, while still at school, of learning a good deal more about the real subject matter out of which modern geometrical systems are built. It is probably easier, and certainly vastly more instructive than a great deal of what they are actually taught… I have not yet encountered a student who finds difficulty with such ideas [projective geometry, the nature of axiom systems, and perspective]… [We must] widen the horizon of knowledge, recognizing, as regards the niceties of logic, sequence, and exposition, that the elementary geometry of schools is a fundamentally and inevitably illogical subject.

G.H. Hardy, from "What is Geometry?" Hardy's Presidential Address to The Mathematical Association in 1925. Reprinted in The Changing Shape of Geometry edited by Chris Pritchard, Cambridge University Press, 2003.

1788

The mathematician is in much more direct contact with reality [than the physicist]. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real'; but a very little reflection is enough to show that the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God: each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense.
… Neither the physicists nor philosophers have ever given any convincing account of what 'physical reality' is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls 'real'. Thus we cannot be said to know what the subject-matter of physics is; but this need not prevent us from understanding roughly what a physicist is trying to do. It is plain that he is trying to correlate the incoherence body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme which he can borrow only from mathematics.
A mathematician, on the other hand, is working with his own mathematical reality… A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy… Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way

We have done no more than to make explicit a few of the instinctive prejudices of the 'mathematician in the street.' Yet with our first demand [mathematical truth is immutable and unconditional] we have antagonized at least two-thirds of the philosophers in the world; and with the second [mathematical theorems are statements about reality] we have reduced our first indiscretion to entire insignificance, since we have committed ourselves, in one form or another, to the objective reality of propositions, a doctrine rejected, I believe, not only by all philosophers, but also by all three of the current schools of mathematical logic.

G.H. Hardy, quoted in Musings of the Masters edited by Raymond G. Ayoub.

1472

Stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation [than mathematics]. A mathematician need not consider himself on the defensive… The public does not need to be convinced that there is something in mathematics.

Chess problems are the hymn-tunes of mathematics. We may learn the same lesson, at a lower level but for a wider public, from bridge, or descending further, from the puzzle columns of the popular newspapers. Nearly all of their immense popularity is a tribute to the drawing power of rudimentary mathematics… What the public wants is a little intellectual 'kick', and nothing else has quite the kick of mathematics.

It would be difficult now to find an educated man quite insensitive to the aesthetic appeal of mathematics… The fact is that there are few more 'popular' subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.

I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, the least different to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside of mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.

There is one comforting conclusion which is easy for a real mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.

If then I find myself writing, not mathematics, but 'about' mathematics, it is a confession of weakness, for which I may rightly be scorned or pitied by younger and more vigorous mathematicians. I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job.

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done… Exposition, criticism, appreciation, is work for second-rate minds.

I believe that mathematical reality lies outside of us and that our
function is to discover, or observe it, and that the theorems which we
prove, and which we describe gradiloquently as our "creations" are simply
notes on our observations.

G. H. Hardy,

156

The elementary theory of numbers should be one of the very best subjects
for early mathematical instruction. It demands very little previous
knowledge; its subject matter is tangible and familiar; the processes of
reasoning which it employs are simple, general and few; and it is unique
among the mathematical sciences in its appeal to natural human curiosity. A
month's intelligent instruction in the theory of numbers ought to be twice
as instructive, twice as useful, and at least ten times as entertaining as
the same amount of "calculus for engineers.''

G. H. Hardy, quoted in Excursions in Calculus, by Robert Young.

157

A mathematician, like a painter or poet, is a maker of patterns. If his
patterns are more permanent than theirs, it is because they are made with
ideas.

Nathaniel Hawthorne, quoted in A Primer of Mathematical Writing by Steven G. Krantz.

740

Life is never so bad at its worst that it is impossible to live; it is never
so good at it's best that it is easy to live.

Gabriel Heatter,

932

The major lesson of all the wars is the fact that nobody takes any lesson out of them.

Hegel,

1394

Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence.

Georg Hegel, from A Dictionary of Quotations in Mathematics by Nowlan

1668

The Nothing Nots.

M. Heidegger, quoted in Sweet Reason, by J. Henle and T. Tymoczko.

159

Problems worthy of attack prove their worth by fighting back.

Piet Hein,

1409

…The definition of irrational numbers, on which geometric representations have often had a confusing influence… I take in my definition a purely formal point of view, calling some given symbols numbers, so that the existence of these numbers is beyond doubt.

Heine, quoted in Analysis by Its History by E. Hairer and G. Wanner.

1112

Where books are burned, in the end people will be burned.

Heinrich Heine, from the U.S. Holocaust Memorial Museum

969

Whenever they burn books they will also, in the end, burn human beings.

Heinrich Heine,

998

I am free because I know that I alone am morally responsible for everything I do.

Robert Heinlein,

844

We have to remember that what we observe is not nature in itself but nature exposed to our method of questioning.

Werner Heisenberg, from Physics and Philosophy.

1059

In my paper the fact the XY was not equal to YX was very disagreeable to
me. I felt this was the only point of difficulty in the whole scheme...and
I was not able to solve it.

W. Heisenberg, from Contemporary Abstract Algebra, by J. Gallian.

160

In this way quantum theory reminds us, as Bohr has put it, of the old wisdom that when searching for harmony in life one must never forget that in the drama of existence we are ourselves both players and spectators. It is understandable that in our scientific relation to nature our own activity becomes very important when we have to deal with parts of nature into which we can penetrate only by using the most elaborate tools.

Werner Heisenberg, from Physics and Philosophy.

1060

In the natural sciences, then, the object of research is no longer nature as such, but a nature confronted by human questions, and in this sense, here too, man encounters himself.

Werner Heisenberg, Quoted in Radical Constructivism: A Way of Knowing and Learning by Ernst von Glasersfeld.

773

We have to remember that what we observe is not nature itself, but nature exposed to our methods of questioning.

Wener Heisenberg, quoted in Teaching as a Subversive Activity, p. 78.

1399

At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at the strangely beautiful interior and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures spread out before me.

Werner Heisenberg, quoted in Insights of Genius: Imagery and Creativity in Science and Art, by Arthur I. Miller.

563

What we observe is not nature itself, but nature exposed to our method of questioning.

Werner Heisenberg,

1283

Orr would be crazy to fly more missions and sane if he didn't,but if he was sane he had to fly them. If he flew them he was crazy and didn't have to; but if he didn't want to he was sane and had to.

Joseph Heller, from Words I Wish I Wrote by Robert Fulgham

950

All good books are alike in that they are truer than if they had really happened
and after you are finished reading one you will feel that all that happened
to you and afterwards it all belongs to you; the good and the bad, the ecstasy, the remorse, and sorrow, the people and the places and how the weather was.

Ernest Hemingway,

999

With the aid of hyperspace philosophy, Theosophy, fantasies like Abbott's Flatland, and the science fiction of Wells and others, the fourth dimension had become almost a household word by 1910. Non-Euclidean geometry never achieved such a widespread popularity, in part because it did not lend itself to such a variety of interpretations. Ranging from an ideal Platonic or Kantian reality -- or even Heaven -- to the answer to all of the problems puzzling contemporary science, the fourth dimension could be all things to all people.

Linda Dalrymple Henderson, from The Fourth Dimension and non-Euclidean Geometry in Modern Art.

663

When understood in their original context, however, "the fourth dimension" and no-Euclidean geometry are far from being the "scourge of every history of modern painting," as they have been termed. Instead, these concepts open the door to our understanding more fully the goals of many seminal artists of the early twentieth century.

Linda Dalrymple Henderson, from The Fourth Dimension and non-Euclidean Geometry in Modern Art.

662

In most sciences one generation tears down what another has built and what
one has established another undoes. In mathematics alone each generation
adds a new story to the old structure.

Herman Henkel, from A Mathematical Journey, by S. Gudder.

161

The struggle to become a better teacher begins all over again with the
advent of each new class.

Martin Henley,

162

The function of education has never been to free the mind and the spirit of
man, but to bind them; and to the end that the mind and spirit of his
children should never escape, Homo Sapiens has employed praise, ridicule,
admonition, accusation, mutilation, and even torture to chain them to the
culture pattern.

The paradox of the human condition is expressed more in education than
elsewhere in human culture, because learning to learn has been and
continues to be Homo Sapiens' most formidable evolutionary task... It must
also be clear that we will never quite learn how to learn, for since Homo
Sapiens is self-changing, and since the more culture changes the faster it
changes, man's methods and rate of learning will never quite keep pace with
his need to learn.

If you do not expect the unexpected, you will not find it; for it is hard to be sought out, and difficult.

Heraclitus, quoted in Mathematics and the Imagination by Edward Kasner and James Newman.

689

I find mathematics an infinitely complex and mysterious world; exploring it is an addiction from which I hope never to be cured.

Reuben Hersh, from The Mathematical Experience

861

The value of a problem is not so much coming up with the answer as in the
ideas and attempted ideas it forces on the would be solver.

I. N. Herstein, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.

166

Very often in mathematics the crucial problem is to recognize and discover
what are the relevant concepts; once this is accomplished the job may be
more than half done.

I. N. Herstein, from Contemporary Abstract Algebra, by J. Gallian.

165

One cannot escape the feeling that these mathematical formulae have an
independent existence and an intelligence of their own, that they are wiser
than we are, wiser even than their discoverers, that we get more out of
them than we originally put into them.

Heinrich Hertz,

167

... I do not consider myself less ignorant than most people. I have been and still am a seeker, but I have ceased to question stars and books; I have begun to listen to the teachings my blood whispers to me. My story is not a pleasant one; it is neither sweet nor harmonious, as invented stories are; it has the taste of nonsense and chaos, of madness and dreams - like the lives of all men who stop deceiving themselves.

Hermann Hesse, from the Prologue of Demain

1062

What Weyl and Brouwer are doing is none other than a revival of Kronecker"s idea. They try to save mathematics by tossing overboard all that provokes concern... They crumble and chop science. If we accepted the reform they propose, then we would run the risk of losing the greatest part of our precious treasure.

David Hilbert, quoted in In Search of Infinity by N.Ya. Vilenkin (translated by Abe Shenitzer).

728

Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure.

David Hilbert, quoted in "Fermat's Last Stand," by Simon Singh and Kenneth A. Ribet, in Scientific American, November 1997.

599

We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can though about things be so much different from things? Can thinking processes be so unlike the actual process of things? In short, can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality?

David Hilbert, quoted in Understanding the Infinite by Shaughan Lavine.

547

The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.

David Hilbert,

1495

This conviction... is a powerful incentive. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no "we shall not know."

David Hilbert, quoted in Bridges to Infinity by Michael Guillen.

535

One must admit that the state we are in now vis-a-vis the paradoxes is in the long run unendurable. Just think of it: in mathematics, this standard of trustworthiness and truth - the forming of concepts of inferences, as learned, taught, and used by all of us, can lead to nonsense. Where is one to find reliability and truth if mathematical thought can fail?

David Hilbert, quoted in In Search of Infinity by N. Ya Vilenkin

1492

[On Cantor's work:] The finest product of mathematical genius and one of
the supreme achievements of purely intellectual human activity.

David Hilbert, quoted in The History of Mathematics, by D. Burton.

170

Mathematics is an organism for whose vital strength the indissoluble union of the parts is a necessary condition.

David Hilber, quoted in Mathematical Thought from Ancient to Modern Times by Morris Kline.

1093

With his theorem, which states that a continuous function of a real variable actually attains its least upper and greatest lower bounds, i.e., necessarily possesses a maximum and a minimum, Weierstrass created a tool which today is indispensable to all mathematicians for more refined analytical or arithmetical investigations.

Hilbert, quoted in Analysis by Its History by E. Hairer and G. Wanner.

1114

In mathematics, as in any scientific research, we find two tendencies
present. On the one hand, the tendency toward abstraction seeks to
crystallize the logical relations inherent in the maze of material that is
being studied, and to correlate the material in a systematic and orderly
manner. On the other hand the tendency toward intuitive understanding
fosters a more immediate group of the subjects one studies, a live rapport
with them, so to speak, which stresses the concrete meaning of their
relations.

David Hilbert, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.

168

Mathematical science is in my opinion an indivisible whole, an organism
whose vitality is conditioned upon the connection of its parts.

David Hilbert, quoted in Excursions in Calculus, by Robert Young.

171

No one will expel us from the paradise that Cantor has created.

David Hilbert, quoted in Patterns in Mathematics, by McCowen and Sequeira.

169

Mathematical science is in my opinion an indivisible whole, an organism
whose vitality is conditioned upon the connection of its parts.

David Hilbert, quoted in Excursions in Calculus, by R.M. Young.

173

For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, juggling with numbers.

David Hilbert, from Geometry and the Imagination

815

The infinite! No other question has ever moved so profoundly the spirit of
man.

David Hilbert,

172

For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers.

David Hilbert, from Geometry and the Imagination.

1242

The conception of the inconceivable [imaginary], this measurement of what
not only does not, but cannot exist, is one of the finest achievements of
the human intellect. No one can deny that such imaginings are indeed
imaginary. But they lead to results grander than any which flow from the
imagination of the poet. The imaginary calculus is one of the masterkeys to
physical science. These realms of the inconceivable afford in many places
our only mode of passage to the domains of positive knowledge. Light itself
lay in darkness until this imaginary calculus threw light upon light. And
in all modern researches into electricity, magnetism, and heat, and other
subtile physical inquiries, these are the most powerful instruments.

Thomas Hill, quoted in Memorabilia Mathematica, by Robert E. Moritz.

175

A pamphlet is never read more than once, but a song is learned by heart and repeated over and over again.

Joe Hill,

1233

The discoveries of Newton have done more for England and for the race, than
has been done by whole dynasties of British monarchs.

Thomas Hill, quoted in The History of Mathematics, by D. Burton.

174

The U.S. Bureau of Labor Statistics predicted that OR [Operations Research]
will be the third fastest- growing career area for U.S. college graduates
from 1990 to 2005. It is also predicted that 100,000 people will be
employed as operations research analysts in the United States by the year
2005.

F. Hillier and G. Lieberman, from Introduction to Operations Research.

Particularly perverse and absurd is the multiple-choice format. I have been doing mathematics now as a professional for nearly 40 years and have never met a situation (outside of finite group theory!) in which I was faced with a mathematical problem and knew that the answer was one of five possibliites. Moreover if faced, artifically, by such a situation, may approach would, and should, be quite different from that in which I simply had to solve the problem.

Tests tyrannize us -- they tyrannize teachers and children. They loom so large that they distort the teaching curriculum and the teacher"s natural style; they occur so frequently, and with such dire consequences, that they appear to the child (and, perhaps, to the teacher) to be the very reason for learning mathematics.

I.J. Good, a wartime colleague and friend, has aptly remarked that it is forunate that the authorities did not know during the war that [Alan] Turing was a homosexual; otherwise, the Allies might have lost the war.

Peter Hilton, from "Cryptanalysis in World War II -- and Mathematics Education," Mathematics Teacher, Oct. 1984.

574

No wonder that Churchill described this effort [the British codebreakers working at Bletchley Park] as "Britian"s secret weapon," a weapon far more effective than the buzz bombs and the rockets that Werner von Braun designed for a German victory, a weapon absolutely decisive, in the judgement of many, in winning the war for the Allies.

Peter Hilton, from "Cryptanalysis in World War II -- and Mathematics Education," Mathematics Teacher, Oct. 1984.

575

Just as any sensitive human being can be brought to appreciate beauty in art, music or literature, so that person can be educated to recognize the beauty in a piece of mathematics. The rarity of that recognition is not due to the "fact" that most people are not mathematically gifted but to the crassly utilitarian manner of teaching mathematics and of deciding syllabi and curricula, in which tedious, routine calculations, learned as a skill, are emphasized at the expense of genuinely mathematical ideas, and in which students spend almost all their time answering someone else's questions rather than asking their own.

In March of 1945, I became a soldier and in April a prisoner of war, surviving in the meadows along the Rhine, always under the open sky in rain and sunshine, scribbling mathematics on toilet paper, the only paper available… To receive ration cards, I had to clean the British barracks. But on the first day, a British soldier asked me in fluent German what I was doing there and what I actually wanted to do. I said, "Mathematics!" He put me in his jeep and drove me home: "Study mathematics!" I did. To this day, I regret that I did not ask his name.

Friedrich E. Hirzebruch,

1513

When we say that anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named.

Thomas Hobbes, quoted in To Infinity and Beyond by Eli Maor.

645

To understand this [Torricelli's Trumpet, a.k.a. Gabriel's Horn] for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.

Thomas Hobbes, quoted in "Torricelli"s Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century," by P. Mancosu and E. Vailati, Isis, vol. 82, 1991.

He told me there was no past. Or should I say, that the past is always on and waiting. Memories coded on a billion chemical loops playing the background like a movie with the sound turned down, until we reach back for that day in time. That face. That kiss. He said, Oh, our marvelous, mysterious little brains. Evolved from a single-celled bacteria we once were, huddling together for survival, and we still communicate as those colonies once did, one impulse colliding with the next impulse, in lightning strikes of fight or flight, leaping the gaps of synapse; and the memories we retrieve most frequently carve the deepest neural pathways, like water eroding a mountain gorge, and become our truth.

Sheri Holman, spoken by the character Eddie Alley in the book Witches on the Road Tonight.

1807

My mind rebels at stagnation. Give me problems, give me work, give me the most abstruse cryptogram, or the most intricate analysis, and I am in my own proper atmosphere. But I abhor the dull routine of existence. I crave for mental exaltation.

Sherlock Holmes, from the character by Sir Arthur Conan Doyle.

1251

The best of a book is not the thought which it contains, but the thought which it suggests; just as the charm of music dwells not in the tones but in the echoes of our hearts.

Oliver Wendell Holmes,

1000

The freeman, casting with unpurchased hand, the vote that shakes the turrets of the land.

Oliver Wendell Holmes,

1367

One's mind, once stretched by a new idea, never regains its original dimensions.

Oliver Wendell Holmes,

1368

A new and valid idea is worth more than a regiment and fewer men can
furnish the former than can command the later.

What a school thinks about its library is a measure of what it feels about education.

Harold Howe,

1001

Fences are for those who cannot fly.

Elbert Hubbard,

1589

When on the brink of complete discouragement, success is discerning that... the line between failure and success is so fine that often a single extra effort is all that is needed to bring victory out of defeat.

What happens to a dream deferred? Does it dry up- Like a raisin in the sun? Or fester like a sore- And then run? Does it stink like rotten meat? Or crust and sugar over- Like a syrupy sweet? Maybe it just sags- Like a heavy load. Or does it explode?

Langston Hughes, (A Dream Deferred).

178

Oh God of dust and rainbows, help us see that without dust the rainbow would not be.

Langston Hughes,

1442

By this means all knowledge degenerates into probability; … There is Algebraist nor Mathematicians so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. Every time her runs over his proofs, his confidence encreases; but still more by the approbation of his friends; and is raised to its utmost perfection by the universal assent and applauses of the learned world. Now 'tis evident that this gradual encrease of assurance is nothing but the addition of new probabilities.

David Hume, from Treatise on Human Nature, quoted in What is Mathematics, Really? by Rueben Hersch, p. 189.

1254

In 1953 I realized that the straight line leads to the downfall of mankind. But the straight line has become an absolute tyranny. The straight line is something cowardly drawn with a rule, without thought or feeling; it is the line which does not exist in nature... Any design undertaken with the straight line will be stillborn. Today we are witnessing the triumph of rationalist knowhow and yet, at the same time, we find ourselves confronted with emptiness. An esthetic void, desert of uniformity, criminal sterility, loss of creative power. Even creativity is prefabricated. We have become impotent. We are no longer able to create. That is our real illiteracy.

Friedensreich Hundertwasser, quoted in The Beauty of Fractals, by H.O. Peitgen and P.H. Richter.

478

Research is formalized curiosity.

Zora Neale Hurston,

1023

The method of fluxions is probably one the greatest, most subtle, and
sublime discoveries of any age; it opens a new world to our view, and
extends our knowledge, as it were, to infinity; carrying us beyond the
bounds that seemed to have been prescribed to the human mind, at least
infinitely beyond those to which the ancient geometry was confined.

Charles Hutton, quoted in Memorabilia Mathematica, by R. Moritz.

179

This universe, I conceive, like to a great game being played out, and we poor mortals are allowed to take a hand. By great good fortune the wiser among us have made out some few of the rules of the game, as at present played. We call them 'Laws of Nature', and honor them because we find that if we obey them we win something for our pains. The cards are our theories and hypotheses, the tricks our experimental verifications. But what sane man would endeavor to solve this problem? …The problem of the metaphysicians is to my mind no saner.

Thomas Huxley, quoted on p. 513 of The Colossal Book of Mathematics by Martin Gardner

1707

Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data.
Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data.

Thomas Huxley,

1594

I warn you I will stop at no point so long as clear reasoning will take me further.

Thomas Huxley,

1593

The known is finite, the unknown infinite; intellectually we stand on an islet in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land.

Thomas Huxley,

1591

The chess-board is the world; the pieces are the phenomena of the universe; the rules of the game are what we call the laws of Nature. The player on the other side is hidden from us. We know that his play is always fair, just and patient. But we know, to our cost, that he never overlooks a mistake, or makes the slightest allowance for ignorance.

Thomas Huxley,

1590

That men do not learn very much from the lessons of history is the most important of all the lessons that history has to teach.

Aldous Huxley, from "See no evil, hear no evil, speak no evil" in Notices of the American Mathematical Society, vol. 45, no. 9, October 1998.

744

The rung of a ladder was never meant to rest upon, but only to hold a man's foot long enough to enable him to put the other somewhat higher.

Thomas Huxley, quoted Essentials of Mathematics by Margie Hale.

955

The greatest tragedy of science - the slaying of a beautiful hypothesis by
an ugly fact.

Thomas Huxley,

180

Sooner or later, false thinking brings wrong conduct.

Julian Huxley,

561

If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood, and many other desirable things. As a man he was a failure; as a monster he was superb.

At least two thirds of our miseries spring from human stupidity, human malice and those great motivators and justifiers of malice and stupidity, idealism, dogmatism and proselytizing zeal on behalf of religious or political idols.

Aldous Huxley,

1379

…You [Leibniz] will not deny that you have discovered a very remarkable property of the circle [pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...], which will forever be famous among geometers.

Huygens, quoted in Analysis by Its History by E. Hairer and G. Wanner.